Trisecting an Angle Using the Cycloid of Ceva

The cycloid of Ceva has the polar equation . To trisect the angle , construct a line parallel to the polar axis (the positive axis). Let be the point of intersection of the cycloid and the line. Then the angle is one-third of the angle . Proof: let angle be and let the point on the axis be such that . Let be the orthogonal projection of on the line . The angle , so . Since , , . So angle equals , but .